Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

Michael Kunzinger, Melanie Graf, Darko Mitrovic

Veröffentlichungen: Beitrag in FachzeitschriftArtikelPeer Reviewed

Abstract

We consider the degenerate parabolic equation
$$
\pa_t u +\Div \mff_{\mx}(u)=\Div(\Div ( A_{\mx}(u) ) ), \ \ \mx \in M, \ \ t\geq 0
$$
on a smooth, compact, $d$-dimensional Riemannian manifold $(M,g)$. Here, for each $u\in \R$, $\mx\mapsto \mff_{\mx}(u)$ is a vector field and $\mx\mapsto A_{\mx}(u)$ is a $(1,1)$-tensor field on $M$ such that $u\mapsto \langle A_{\mx}(u) \mxi,\mxi \rangle$, $\mxi\in T_\mx M$, is non-decreasing with respect to $u$. The fact that the notion of divergence appearing in the equation depends on the metric $g$ requires revisiting the standard entropy admissibility concept. We derive it under an additional geometry compatibility condition and, as a corollary, we introduce the kinetic formulation of the equation on the manifold. Using this concept, we prove well-posedness of the corresponding Cauchy problem.
OriginalspracheEnglisch
Seiten (von - bis)4787-4825
FachzeitschriftJournal of Differential Equations
Jahrgang263
Ausgabenummer8
PublikationsstatusVeröffentlicht - 2017

ÖFOS 2012

  • 101002 Analysis

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